by wendy » Mon May 04, 2009 12:17 pm
All space is indeed curved, where 'flat' simply means 'curved at the same measure of all-space'.
There are some things to be noted here. Firstly, space is not curved in anything: it is a property of space itself. It has nothing to do with space-time, or the device where they show black holes in deep pits, but rather a geometric thing.
The thing where one has balls sitting in pits on a surface, is a kind of representation of space, where downwards gravity effects real curvature. It has time, (ie time is not one of the coordinates), since it's intended to be a billiard table, where the slope and speed traces out a trajectory of a thing.
In geometry, curvature has to do with the perimeter of a circle, when compared with its length. For homogeneous isotropic spaces, the sort of thing that we talk about with hyperbolic, euclidean and spherical geometry, this measure depends the measure of the radius against a universal radius. For real space, it varies in direction and place.
Were ye to take a circle, whose circumference is say, 2.pi, the diameter gives a chord of 2. However, the chord from the centre of the circle to its perimeter can be greater or lesser than one. When it's less than 1, the curvature is positive, and greater than 1, it's negative. Zero curvatrure gives euclidean space, but this is zero-curvature, is not necessarily flat.
A curve can be enbedded in a curve of lesser curvature (or greater radius). Ye can draw a circle of diam 20 inches on a sphere of diameter 30 inches, or on the plane, or on a sphere of 20 inches. You can't do it on a 10-inch sphere. On the 20-inch sphere, the circle is straight (that is, it bisects the length of the circuferences at each point). You can't draw a euclidean line on any sphere, but ye can draw one on a hyperbolic plane. It's horridly a crooked line. The horizon of the hyperbolic space is not a straight line, but is very bent.
None the same, a Euclidan plane, of curvature zero, can be represented by a polytope like {3,6} or {4,4}. In euclidean space, this is flat. since all-space is zero curvature. So we see for example, a polytope {4,4} is a tiling of squares, with half-space as interior. In hyperbolic space, the thing has angles less than 180°, can for example have angles like 120° or even 90°. It's still zero-curvature, but it's not flat.
When it is said that one can do away with gravity, by making space curved at any point, the thing is that one supposes at a point P, that different degrees of arc have different lengths, and that space is in tension according to the length. Where space is negatively curved, the length of each degree nearer the mass has a greater length, and a body at rest is pulled towards this direction more than others: it falls in the direction where the curvature is least: ie the degrees are longer. Likewise, a straight line bisects the circumference at each point, will tend to vere towards a mass, because points near the mass are longer than those opposite, and the half-circuference is less than 180°.